Bug Trophy - Catch the largest bug by the end of the Bug Contest.
Fish Trophy - Catch the largest fish by the end of the Fishing Contest.

Fishing rod
Talk to Tortimer the mayor during the Fishing Contest to get a free fishing rod.

Bug net
Talk to Tortimer the mayor during the Bug Catching Contest to get a free bug net

Mush Furniture set
Look under the various mushrooms that will appear in your town. They usually appear around 3:00 p.m. daily, in front of trees. You may find an item from the Mush Furniture set (which includes furniture, carpet, wallpaper, and appliances). You can sell unwanted pieces at Tom Nook's store.

Tortimer: Furnishings
Talk to Tortimer on the following holidays to get the corresponding item:

Kicks: Change shoe color
Find Kicks, the skunk, sitting on a set of stairs. Have him shine your shoes and their color will now match your hair. Note: You can change your hair color at The Shampoodle. Select "Change style" to return your shoes to their original color.

Unlock Harvest Set
Talk to Tortimer after 3pm on Harvest Day, (which this year was Nov. 27). He will give you a Fork and Knife pair, telling you that the 'Special Guest' hasn't shown up yet. There will be a Turkey character, Franklin, hiding somewhere in your town. You will be asked to trade himm the Fork & Knife. If you do, he'll give you an item from the Harvest Set.

Unlock Pirate Ship Theme
If you see Pascal in your town, give him a scallop(Not a white scallop. That doesn't work) and he will give you a random Pirate Ship theme item. Do not talk to him until you have the scallop with you or he swims away.

Unlock Spooky Series
Every Halloween starting at 6:00 p.m., Jack will be at a random in your town. The only problem is that everyone dresses up like him! However, you'll notice which one is the real Jack simply by the fact that he won't follow you around, unlike the villagers. Make sure that you have candy before you talk to him, or else he will play a trick on you. If you give him candy, he'll give you an item from the "Spooky Series". There are 12 in total.